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The University of Sydney

School of Mathematics and Statistics

Tutorial 2 (Week 9)

MATH2069/2969: Discrete Mathematics and Graph Theory Semester 1, 2012

More difficult questions are marked with either * or **. Those marked * are at the level 

which MATH2069 students will have to solve in order to be sure of getting a Credit, or to

have a chance of a Distinction or High Distinction. Those marked ** are mainly intended 

 for MATH2969 students.

1.   Determine whether each of the following sequences is the degree sequence of agraph. If so, draw a picture of such a graph.

(a) (2, 2, 2)

(b) (0, 1, 2, 3)

(c) (2, 3, 3, 4, 4, 5)

(d) (2, 3, 4, 4, 5)

(e) (1, 1, 1, 1, 4)

(f) (1, 3, 3, 3)

(g) (1, 2, 2, 3, 4, 4)

(h) (1, 3, 3, 4, 5, 6, 6)

2.   Suppose that  G  has 23 vertices, and every vertex has degree at least 11.

(a) Is it possible that  G is regular of degree 11?

(b) Prove that G is connected.

3.   For each of the following graphs, either find an Eulerian circuit, find an Euleriantrail, or explain why neither exists.

(a)

a

b

cd

e   f 

(b)   a

b

cd

e  f 

g

hi

 j

(c)a

b d

c

e   f 

*(d)

ab

c

d

e

f g  h

i

 j

k

l

4.   Find the number of vertices, the number of edges, and the degrees of:

(a) the complete bipartite graph K  p,q ; (b) the cube graph  Qm.

*5.  What is the smallest number   n   for which there exist two non-isomorphic graphswith  n  vertices having the same degree sequence?

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6.   Determine whether the graphs represented by the following pictures are Hamilto-nian or not. Explain your answers.

(a)

(b)

(c)

*(d)

*(e)

*(f)

7.   Show that a Hamiltonian graph cannot contain a bridge.

8.   A club plans a series of dinners in which the members will be sitting around around table. They want the seating plans for these dinners to have the propertythat every member sits next to two people they have never sat next to before. Forinstance, if there are five members, they can hold two dinners with the followingseating plans:

1

2

34

5

1

3

52

4

After these two dinners ever member has sat next to the other four, so there cannotbe any further dinners.

(a) If there are 7 members, find the maximum possible number of dinners theclub could hold.

**(b) Repeat the previous part where 7 is replaced by any odd number  n.

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